Is there a term or classification for words (non-adjectives, for lack of a better classifier) that are combined with hyphens that serve the purpose of an adjective? For example, "run-of-the-mill applicant" or "less-than-ideal scenario." Are there any specific rules that dictate how these word groupings are handled?
They are called compound adjectives. You will find an overview with some rules and examples on Wikipedia. Note that you should normally use hyphens ( - ), not dashes ( – or — ) in compound adjectives.